WEBVTT
00:00:01.250 --> 00:00:08.020
Simplify sin 𝜃 plus the cos of 270 degrees plus 𝜃.
00:00:09.040 --> 00:00:19.250
In order to simplify this expression, we will begin by finding an equivalent expression to the cos of 270 degrees plus 𝜃.
00:00:20.010 --> 00:00:23.900
One way of doing this is by considering the unit circle.
00:00:24.770 --> 00:00:35.200
Adding 270 degrees to 𝜃 puts us in the same position on the unit circle as subtracting 90 degrees from 𝜃.
00:00:36.000 --> 00:00:44.230
The cos of 270 degrees plus 𝜃 is equal to the cos of 𝜃 minus 90 degrees.
00:00:44.860 --> 00:00:55.040
This is similar to one of our cofunction identities, which states that the cos of 90 degrees minus 𝜃 is equal to sin 𝜃.
00:00:55.900 --> 00:01:06.340
If we factor negative one out of the expression in our parentheses, we have the cos of negative 90 degrees minus 𝜃.
00:01:07.310 --> 00:01:14.650
As cosine is an even function, the cos of negative 𝛼 is the same as the cos of 𝛼.
00:01:15.400 --> 00:01:21.500
This means that our expression is the same as the cos of 90 degrees minus 𝜃.
00:01:22.180 --> 00:01:27.840
We can then use the cofunction identity so that this is equal to sin 𝜃.
00:01:28.660 --> 00:01:39.780
Replacing the cos of 270 degrees plus 𝜃 with sin 𝜃, our original expression becomes sin 𝜃 plus sin 𝜃.
00:01:40.470 --> 00:01:43.840
This is equal to two sin 𝜃.
00:01:44.860 --> 00:01:51.240
An alternative method would have been to have used the sum identities or addition formulae.
00:01:52.090 --> 00:02:01.860
One of these states that the cos of 𝛼 plus 𝛽 is equal to cos 𝛼 cos 𝛽 minus sin 𝛼 sin 𝛽.
00:02:02.630 --> 00:02:09.720
Considering our expression, we will let 𝛼 be 270 degrees and 𝛽 be 𝜃.
00:02:10.710 --> 00:02:22.250
This gives us cos of 270 degrees multiplied by cos 𝜃 minus sin of 270 degrees multiplied by sin 𝜃.
00:02:23.090 --> 00:02:31.610
The cos of 270 degrees is zero, and the sin of 270 degrees is negative one.
00:02:32.450 --> 00:02:44.190
Our expression simplifies to zero multiplied by cos 𝜃 minus negative one multiplied by sin 𝜃, which is equal to sin 𝜃.
00:02:45.070 --> 00:02:49.230
This confirms the expression we got using our first method.
00:02:50.380 --> 00:02:58.210
sin 𝜃 plus the cos of 270 degrees plus 𝜃 is equal to two sin 𝜃.